Solutions for Session 9, Part C

See solutions for Problems: C1 | C2 | C3 | C4

 Problem C1 On most pineapples, all three numbers will be consecutive Fibonacci numbers: 8, 13, and 21.

 Problem C2 The ratios seem to be approaching one number, which is about 1.618, to three decimal places.

 Problem C3 If the pattern continues, this ratio should be fairly close to the ratios found in the table in Problem C2; it should also be very close to the ratio of the other consecutive Fibonacci numbers around it.

 Problem C4 Consider the ratio of sides in each golden rectangle. In Rectangle 1, the ratio is ø to 1. In Rectangle 1 + 2, the ratio is ø + 1 to ø. These are similar rectangles, so the ratios must be equal: Cross-multiplying and simplifying the equation gives us a quadratic equation: ø2 - ø - 1 = 0. This equation does not factor, so we must use the quadratic formula to find the value of ø; the two possible values are . Since the side of a rectangle can't be negative, . Evaluating this on a calculator gives us the decimal 1.618 to three decimal places, as expected.